Mathematical Sciences: Crystalline Variational Problems

数学科学：晶体变分问题

• 批准号: 8803395
• 负责人: Jean Taylor
• 金额: $ 9.62万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8803395&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Crystalline; Variational; Problems

Jean Taylor will continue her work on surfaces of minimum energy. She will focus her attention on minimizing energy functionals which do not possess directional uniformity, this is known as anisotropism. The techniques to be used come from a branch of the calculus of variations, geometric measure theory, which is having an ever increasing impact on many areas of geometric analysis. This work has applications in materials research and some joint projects with John Cahn, a metallurgist, will be continued. One of her main objectives will be to understand the geometric consequences of minimization of crystal- type anisotropic surface energies. In a slightly different direction she will try to develop and employ combinatorial techniques of area minimization. These latter techniques become appropriate in the polyhedral situations which frequently arise. The energy functionals have an integral representation inwhich the integrand, known as the free energy, is defined on the unit sphere. The free energy in turn gives rise to the Wulff shape, which is the obect of most of Taylor's studies. She will study several new types of singularities in these surfaces and some new rules for orientations in heterogeneous nucleation, which have been predicted in some of her earlier work with Cahn. Some of this will involve allowing edge energies in the integral being minimized. Other investigations will study the evolution of the minimizing surfaces as the energy function is allowed to vary.

让·泰勒（Jean Taylor）将继续在最低能量的表面上工作。她将注意力集中在最大程度地减少没有方向均匀性的能量功能上，这被称为各向异性。所使用的技术来自变化的计算的分支，几何措施理论，该理论对几何分析的许多领域都有越来越多的影响。这项工作在材料研究中有应用，并将继续与冶金学家约翰·卡恩（John Cahn）进行一些联合项目。她的主要目标之一是了解最小化晶体各向异性表面能的几何后果。在稍微不同的方向上，她将尝试开发和采用最小化面积的组合技术。这些后一种技术在经常出现的多面体情况下变得合适。 能量函数具有积分表示，在单位球体上定义了被称为自由能的整数。自由能反过来产生了沃尔夫的形状，这是泰勒大多数研究的肥胖。她将研究这些表面中的几种新型奇点，并研究一些新的关于异质成核方向的新规则，这些规则已在她与CAHN的一些早期工作中得到了预测。 其中一些将涉及允许在积分中最小化的边缘能量。随着允许能量函数的变化，其他研究将研究最小化表面的演变。